Answer:
a) Wavelength of electron = 8.21 × 10⁻¹¹ m = 821 pm
Wavelength of proton = 4.50 × 10⁻¹³ m = 0.45 pm
The electron has a higher wavelength than the electron.
b) Wavelength of electron = 4.61 × 10⁻¹⁰ m = 461 pm
Wavelength of proton = 1.30 × 10⁻¹³ m = 0.130 pm
The electron once again has a higher wavelength.
Note 1 pm = 10⁻¹² m
Explanation:
a) The relationship between wavelength, mass of a particle and its speed is given in the De Broglie's equation
λ = h/mv
For the two particles
h = Planck's constant = 6.63 × 10⁻³⁴ J.s
v = 8.83 × 10⁶ m/s
For an electron
m = 9.11 × 10⁻³¹ kg
Wavelength = (6.63 × 10⁻³⁴)/(9.11 × 10⁻³¹ × 8.83 × 10⁶) = 8.21 × 10⁻¹¹ m = 821 pm
For a proton
m = 1.67 × 10⁻²⁷ kg
Wavelength = (6.63 × 10⁻³⁴)/(1.67 × 10⁻²⁷ × 8.83 × 10⁶) = 4.50 × 10⁻¹³ m = 0.45 pm
b) The velocity needs to be first obtained from the kinetic energy relation. Before using the De Broglie's equation.
K.E = mv²/2
v = √(2K.E/m)
For the electron,
m = 9.11 × 10⁻³¹ kg
v = √[(2 × 7.81 × 10⁻¹⁵)/(9.11 × 10⁻³¹)]
v = 1.31 × 10⁸ m/s
Wavelength = h/mv
Wavelength = (6.63 × 10⁻³⁴)/(9.11 × 10⁻³¹ × 1.31 × 10⁸) = 4.61 × 10⁻¹⁰ m = 461 pm
For the proton,
m = 1.67 × 10⁻²⁷ kg
v = √[(2 × 7.81 × 10⁻¹⁵)/(1.67 × 10⁻²⁷)]
v = 3.06 × 10⁶ m/s
Wavelength = h/mv
Wavelength = (6.63 × 10⁻³⁴)/(1.67 × 10⁻²⁷ × 3.06 × 10⁶) = 1.30 × 10⁻¹³ m = 0.130 pm
